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Midterm Exam
Lasers and Modern Optics
Professor Duncan MacFarlane
Thursday, October 26, 2006 – Tuesday, November 7, 2006
Instructions: Work through the following problem neatly and professionally, and
without collaboration of any kind. You may consult your own notes, and any
published textbook. Please be neat and professional in your presentation. Include
plots and words to expand on your results. A perfect score on this exam is 40
points.
1. Beam Optics and Laser Resonators (10 points)
Consider the following laser resonator comprising a curved mirror (radius Rc)
and a flat mirror separated by a length (Lc). The flat mirror is the output coupler
from which the laser beam emerges.
Rc
Lc
a. Analyze this cavity and derive the ranges for Rc and Lc that support a
Gaussian beam.
b. Pick one of these stable resonators (choose numbers that satisfy the
condition for part a) and solve for the mode of this cavity. Specifically,
plot the spotsize, w(z), and the radius of curvature, R(z), of the mode
everywhere in the cavity.
c. Put a focusing lens immediately after the output coupler of the resonator
as shown below. Design the lens to minimize the focused spot size:
i.
What is the minimum spot produced by this configuration?
ii.
Where is the minimum spot size located?
iii.
iv.
Does the radius of curvature go flat at this same point?
The electric field is stronger at the focus of the lens than
immediately out of the laser. What is this electric field
enhancement factor?
Rc
Lc
f
2. Laser Diodes (10 points)
For lasers ( 100-um wide apertures, w = 100 um) with uncoated facets, R = 0.3,and
operating at λ= 0.808 um, the following data was obtained from length studies.
L(cm)
α(1/cm)
Ith(A)
Jth(A/cm2)
q
η d [( dP
dI ) hν ]
0.1
0.15
0.2
12
8
6
0.311
0.392
0.478
311
261
239
0.56
0.5
0.45
(a) Find the internal efficiency, ηi, and internal loss, αi.
(b) Find the modal gain coefficient, Γg0, and the transparency current, Jtr. Use
the logarithmic approximation for the gain, g = g 0 ln (ηi J J tr ) . The gain is in terms
the current density, J.
(c) For a Power level of P = 1 W, graphically determine the device length, L, that
provides the minimum operating current, I.
3. Quantum States (10 points)
Solve the steady state Schrodinger wave equation for an electron in an infinite
well of length a (particle in a box).
− ih 2 ∂ 2 Ψ
+ (V − E )Ψ = 0
2m ∂x 2
Inside the well, V=0, outside the well, V=∞
Show that the boundary conditions demand quantized energy states, n=1, 2, 3
with energies En given by:
h2n2
En =
8ma 2
4. FM Microchip Laser (10 points)
Consider the following laser comprised of a sliver of Er:YVO4 epoxied to
Lithium Niobate (LiNb03), and pumped by a laser diode. The Er doped crystal is
the gain medium. Lithium Niobate has an index of refraction that changes with
voltage.
a. What is the approximate output wavelength of the laser?
b. What is the longitudinal mode spacing of the laser if there is no voltage
applied to the lithium niobate?
c. Derive an equation to describe what happens to the lasing frequency as a
voltage signal is put on the lithium niobate.
d. Comment on the utility of this laser.